On the homotopy of finite CW-complexes with polycyclic fundamental group

Abstract

Let X be a finite CW-complex of dimension q. If its fundamental group ${\pi }_1(X)$ is polycyclic of Hirsch number h>q we show that at least one of the homotopy groups ${\pi }_i(X)$ is not finitely generated. If h=q or h=q-1 the same conclusion holds unless X is an Eilenberg-McLane space $K({\pi }_1(X),1)$.

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